Place all the marinade ingredients into a large bowl, and whisk to combine. Slice the chicken lengthwise into ~ 1/4 inch slices. Lay the slices flat on your cutting surface and pound any uneven areas, so that the strips are of uniform thickness.
Add the chicken strips to the marinade, and toss to coat.
Cover and let marinate in the refrigerator for at least 30 minutes.
If using wooden skewers, now is a good time to soak them in water before grilling.
For the cucumber relish
Add the sugar and vinegar to a small bowl, and whisk to combine until the sugar dissolves.
Add cucumber, shallot, cilantro, and chile (if using) and stir to coat. Set aside.
For the peanut sauce
Place all peanut sauce ingredients into a medium bowl, and whisk to combine. Set aside.
To cook the chicken
Brush your grill with oil – fire to medium-high heat. Thread your chicken strips onto the skewers, so that they will lay as flat as possible.
Place chicken skewers on the grill, and cook until internal temperature reaches 165° F, flipping halfway through (~3 minutes for each side).
Remove from grill, and assemble your lettuce wraps.
To assemble lettuce wraps
Lay the lettuce leaves out on a serving platter. Remove a chicken strip from the skewer, and place inside the lettuce leaf. You may need to cut the strips in half if they are longer than your lettuce.
Add a dollop of peanut sauce, a bit of the cucumber relish, and garnish with mint, cilantro, carrots, peanuts, Thai chiles, and a squeeze of lime.
Grab the wrap like a taco, and devour! Best served while the chicken is still warm.